6
$\begingroup$

This is the definition of spin structure according to Wikipedia:

enter image description here

which is supposed to be the standard definition. But in the book The Geometry of Four-Manifolds (Donaldson-Kronheimer, page 76) one finds a rather different definition, at least for a 4-dimensional vector space ($S$ is supposed to be a general two-dimensional complex vector space with Hermitian metric and compatible complex symplectic form):

enter image description here

What is the meaning of the second definition? Everything seems quite involved and unrelated. Any idea will be helpful and welcome.

$\endgroup$
2
  • $\begingroup$ Take a look also at mathoverflow.net/questions/122748/what-is-a-spinor-structure and mathoverflow.net/questions/66681/… It doesn't make much sense to me to define a spin structure on a single vector space, like in your second definition. We should really consider vector bundles. $\endgroup$ Dec 6, 2014 at 10:30
  • $\begingroup$ The Donaldson-Kronheimer definition gathers in it all the facts about a spin structure that are needed in gauge theory. They all follow from the representation theory of Spin(4). In particular, they are needed to define the Dirac operator. $\endgroup$ Dec 6, 2014 at 10:37

2 Answers 2

10
$\begingroup$

As it stands, the second definition is a concrete description of the spin group in dimension four. It defines an action of the simply connected group $SU(2)\times SU(2)$ on a vector space of real dimension four, which preserves a positive definite inner product, and this identifies $SU(2)\times SU(2)$ with the universal covering of the special orthogonal group of this four-dimensional Euclidean space. To view it as a definition of a spin-structure on a manifold, one has to do this in each point of the manifold. This means that the spin strucuture in this sense is given by two auxiliary complex rank two bundles $S^+$ and $S^-$ which are endowed with Hermitian bundle metrics and compatible complex symplectic forms, togehter with an isomorphism between the tangent bundle and the bundle $Hom_J(S^+,S^-)$ which respects the inner products on the two spaces (the given Riemannian metric and the inner product constructed point-wise as in definition 2). The equivalence between the two definitions is obtained as follows: To go from definition 1 to definition 2, you form the associated bundles corresponding to the two basic (complex) spin-representations of $Spin(n)$. In the opposite direction, you form the frame bundle of $S^+\oplus S^-$ (with structure group $SU(2)\times SU(2)$, which is isomorphic to $Spin(n)$) and the identification of $Hom_J(S^+,S^-)$ with $TM$ preserving bundle metrics shows that this bundle is a two fold covering of the $SO(n)$-frame bundle associated to the Riemannian metric.

$\endgroup$
4
  • 1
    $\begingroup$ Which is the explicit definition of the action of $SU(2)\times SU(2)$ on $V$ given the isomorphism $\gamma:V\longrightarrow Hom_J(S^+,S^-)$? $\endgroup$
    – Jjm
    Jan 16, 2015 at 10:02
  • 1
    $\begingroup$ You just take $V$ to be the space $Hom_J(S^+,S^-)$. This is a real vector space of dimension $4$ endowed with an inner product, and the action of $SU(2)\times SU(2)$ on that space by construction preserves that inner product. Therefore you get a homomorphism to $SO(V)$ and you have to verify that this is a two-fold covering. $\endgroup$ Jan 16, 2015 at 11:11
  • 1
    $\begingroup$ But how does $SU(2)\times SU(2)$ explicitely act on $Hom_J(S^+,S^-)$? $\endgroup$
    – Jjm
    Jan 16, 2015 at 11:14
  • 2
    $\begingroup$ Take the first copy of $SU(2)$ as quaternionic automorphisms of $S^+$ and the second as quaternionic automorphisms of $S^-$. Then the action is given by composition from both sides, i.e. $((A,B)\cdot f)(x)=B(f(A^{-1}x))$. $\endgroup$ Jan 16, 2015 at 11:35
2
$\begingroup$

As Liviu says, these properties follow from the usual definition of spin structures (in dimension 4). It's a little more work to prove that the existence of these bundles with Clifford multiplication gives a spin structure in the usual sense. Details are given in the Kronheimer-Mrowka book Monopoles and 3-manifolds. They do spin^c structures there, which are even more useful.

The discussion is a little easier in dimension 3 (op. cit.). I don't know if there is an analogous way of defining spin and spin^c structures in other dimensions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.